Quadrupole mass spectrometers are well known in the art. A conventional mass spectrometer, shown in FIG. 1, includes an ion source 1 for forming a beam of ions 2 of the sample to be mass analyzed, a quadrupole filter which comprises two pairs of cylindrically or preferably hyperbolic rods 3 arranged symmetrically about a central axis and positioned to receive the ion beam. A voltage source 4 supplies r.f. and DC voltages to the rods to induce a substantially quadrupole electric field between the rods. An ion detector 5 detects ions which pass longitudinally through the rods from the ion source to the detector. The electric field causes the ions to be deflected or oscillate in a transverse direction. For a particular r.f. and DC field, ions of a corresponding mass-to-charge ratio follow stable trajectories and pass through the quadrupole and are detected. Other ions are caused to deflect to such an extent that they strike the rods. The apparatus serves as a mass filter. The operation of quadrupole mass filters is described in Paul, et al. U.S. Pat. No. 2,939,952.
In one mode of operation, the mass spectrometer is operated as a narrow pass filter in which the r.f. and DC voltages are selected to pass a single mass or a range of masses. In another mode of operation, the quadrupole is operated with r.f. only. The voltage of the r.f. is scanned to provide at the detector a stepped output such as shown in FIG. 2. If the r.f. voltage is increased, ions of consecutively higher mass are rejected and the ion current at the detector reduces in steps as shown in FIG. 2. Differentiation of the steps provides a mass spectrum.
In order to provide a basis for a better understanding of this invention, a theoretical explanation of the operation of a quadrupole mass filter is provided. The voltages applied to the rods set up a quadrupole field between the rods. In a quadrupole field the force on a charged particle is proportional to its displacement from the central axis or point. In the context of the present discussion, only the case for a two-dimensional electrostatic field is relevant. A two-dimensional field can be formed by four cylindrical, or preferably hyperbolic, electrodes arranged symmetrically about a central axis as described in U.S. Pat. No. 2,939,952 and shown in FIG. 1.
Opposing electrodes are connected in pairs, and the coordinate system used to describe the structure places one pair of rods on the xz plane and the other pair on the yz plane, with z as the central longitudinal axis. A voltage 2U is differentially applied to the pairs of rods such that one rod pair has a potential U and the other rod pair has a potential -U. This voltage can be an ac and/or a DC voltage. The ac voltage oscillates at a frequency f, which has units of cycles per second or hertz (Hz). The frequency can also be expressed in units of radians per second (.omega.) by the relationship .omega.=2.pi.f. In practice this frequency is within the radio frequency, r.f., domain and so is generally referred to as the r.f. frequency. The radius of a circle inscribed within the hyperbolic electrode structure is r.sub.o. The containment fields are described by equation (1). ##EQU1## U.sub.DC is the constant potential difference between the pairs of electrodes and U.sub.r.f. is the peak value of the time-varying portion of the potential difference between the pairs of electrodes. The frequency of the time varying portion of the field is .omega., which is expressed as radians per second and the term cos (.omega.t) fixes the phase as zero at t.sub.o. Taking derivatives with respect to x and y yields equations (2) and (3) which express the field gradient in the independent dimensions. ##EQU2## In each dimension, the force exerted upon a charged particle is the product of the negative of the field gradient, d.PHI./dx or d.PHI./dy as expressed above, and the charge e. From Newton's laws it is known that force equals mass times acceleration, as in equation (4). Acceleration is d.sup.2 x/dt.sup.2 for the x dimension and d.sup.2 y/dt.sup.2 for the y dimension; therefore, equation (4) can be rewritten for the independent x and y dimensions as equations (5) and (6). ##EQU3## Equations (5) and (6) can be expanded and rearranged to give equations (7) and (8) which fully describe the motion of a particle under the influence of a quadrupole field. ##EQU4## The only distinction between the two equations for the orthogonal planes is the change in sign, which for the ac component represents a 180 degree phase shift in the applied voltage. The motion in the xz plane is independent of the motion in the yz plane only in the sense that the motion described by the first equation is a function of the x displacement, and the motion described by the second equation is a function of the y displacement. The phase relationship to the containing field however is important, as will be seen later.
These equations of motion are differential equations of a type known as the Mathieu equation. Substitution of the definitions of .xi., q.sub.u and a.sub.u, as shown by equations (9), (10) and (11) into the two equations of motion, equations (7) and (8), converts them into a standard form of the Mathieu equation shown in equation (12). Here the dependent variable u can be considered a generalized term for displacement, representing either the x or the y displacement. The parameter .xi. can be considered as a normalized unit of time such that .xi. increases by .pi. for each cycle of the r.f. field. ##EQU5##
The solutions to the Mathieu equation have been extensively characterized. Since the Mathieu equation is a linear differential equation its general solution will be a linear combination of two independent solutions. Equation (13) is one representation of the general solution of the Mathieu equation. ##EQU6##
The general solution is either stable or unstable depending upon whether the value of u(.xi.), which represents a particle's transverse displacement, remains finite or increases without limit as .xi. or time approaches infinity. This depends upon the parameters a.sub.u and q.sub.u, which in turn are functions of the mass-to-charge ratio of the particle, the quadrupole dimensions, the amplitude of the applied voltages and the frequency of the r.f. voltage.
The answer to the question of the stability of an ion's trajectory lies in the parameter .mu.. It can be shown that only for the case where .mu. is purely imaginary, so that .mu.=i.beta., where .beta. is real and not a whole number, will the solution be stable. Using Euler's identities, the complex exponential expression for such a stable solution can be rewritten as equation (14). ##EQU7## In this solution, n is an integer and A and B are constants of integration, which depend upon the initial conditions of position and velocity of the ion in the u dimension.
The combinations of a.sub.u and q.sub.u which yield Mathieu equations producing stable trajectories (solutions) can be described graphically by what is called a stability diagram. FIG. 3 is such a diagram. The coordinates of this diagram are the parameters of the Mathieu equation, a.sub.u and q.sub.u. The shaded regions represent combinations of a.sub.u and q.sub.u which correspond to Mathieu equations yielding unstable trajectories. The unshaded regions therefore represent combinations of a.sub.u and q.sub.u which correspond to Mathieu equations which yield stable trajectories.
The above discussion of the Mathieu equation and the character of its solutions started with the demonstration that the two differential equations representing the transverse motion of an ion in transit through a quadrupole mass filter were, in fact, Mathieu equations. As stated above, the terms stable and unstable refer only to whether the ion's trajectory, u(.xi.), is bounded or unbounded as time or .xi. approaches infinity. For an ion to transit the mass filter without striking one of the electrodes, the equations of motion in each transverse dimension must correspond to stable motion; that is, the solutions to the equations of motion for both the x and y dimensions must be characterized as stable. The importance of such combined stability leads to construction of a combined stability diagram which characterizes the stability of the solutions of both equations of motion. Such a combined stability diagram is obtained by overlaying stability diagrams representing each equation of motion on a common coordinate system. The stability diagram for the equation of motion in the y dimension when plotted on the a.sub.x, q.sub.x coordinate system (rather than the a.sub.y, q.sub.y coordinate system), is identical to FIG. 3, except that it is turned upside down, as the horizontal and vertical axes are inverted. When such a diagram is overlaid with the stability diagram for motion in the x dimension a combined stability diagram is produced. The areas of overlap between x and y stability indicate regions of combined stability and relate to operating conditions that would allow ions to transit a mass filter. There are a number of areas where regions of x and y stability overlap. With respect to the operation of quadrupole mass filters, the large region of combined stability positioned on the q.sub.x axis ranging between q.sub.x =0 and ca. 0.908 is the main region of interest. FIG. 4 is an enlarged view of this region of the combined stability diagram. Located further out on the q.sub.x axis, at a q.sub.x of ca. 7.5, is another small area of combined stability which is of some importance.
A distinction must be made between ions having stable trajectories which do not exceed the inner dimensions of the electrode structure and those which do. Combined stability can be considered a necessary but not a sufficient condition for transit through a quadrupole mass filter. Ions enter a quadrupole with a finite axial (z dimension) velocity and exit after a time, t.sub.exit, which depends upon the length of the device and the axial velocity. If the ion's initial transverse displacements and velocities (initial conditions) upon entry into the quadrupole field, in combination with the parameters a.sub.x and q.sub.x, specify a trajectory which achieves a displacement any greater than r.sub.o in less than the transit time, t.sub.exit, then the ion will most likely strike an electrode and be lost. Transit depends therefore on the combination of advantageous ion entry position and velocity, generally referred to as initial conditions, as well as the stability of the ions trajectory within the quadrupole field. It should be noted that in real quadrupoles, ions having trajectories that are unstable may sometimes transit the quadrupole. This occurs only when transit times are short, initial conditions are favorable and when the a.sub.u and q.sub.x for the ion correspond to a point just outside of a boundary of a combined stable region.
Stable ion trajectories can be further characterized by their characteristic frequencies. Inspection of equation (14) reveals that a stable trajectory can be expressed as an infinite series of sinusoidal terms. The frequencies of all of these terms are defined by the main r.f. frequency, and the characteristic frequency parameter, .beta..sub.u. It should be noted that in the normalized time units of the standard Mathieu equation, the frequency of the r.f. component of the quadrupole field is always 2, and that .beta..sub.u can be interpreted as a normalized frequency. .beta..sub.u is a function of a.sub.u and q.sub.u only. This functionality is expressed in FIG. 3 as iso-.beta..sub.x and iso-.beta..sub.y lines. For the combined region shown in FIG. 4, which concerns practical mass spectrometry, both .beta..sub.x and .beta..sub.y are zero (0.0) at the origin (a.sub.x =0.0, q.sub.x =0.0). For both the x and y dimensions, the parameter .beta..sub.u increases to 1.0 at q.sub.x =0.908 along the a.sub.u =0.0 axis. At the upper apex of the stability area shown in FIG. 3, a.sub.x =0.237 and q.sub.x =0.706, .beta..sub.x =1.0 and .beta..sub.y =0.0.
From equation (14) it can be shown that for any value of .beta..sub.u, the terms cos (2n+.beta..sub.u).xi. and sin (2n+.beta..sub.u).xi. described a set of characteristic frequencies spaced plus or minus (1-.beta..sub.u).omega./2 from 1/2.omega., 11/2.omega., 21/2.omega. etc. This frequency pattern may be expressed in terms of .omega. or f. The ion's motion is a mixture of these frequencies with each contributing according to the magnitude of the coefficients, C.sub.2n.
The coefficients, C.sub.2n, are functions of only a.sub.u and q.sub.u. As n proceeds from zero in either the positive or negative direction, the magnitudes of the coefficients, C.sub.2n, decrease. N. W. McLachlan (Philosophical Magazine 36 [1945] pp 403-414) describes a method that allows the computation of these coefficients by arbitrarily choosing the highest subscript to be evaluated and then solving for all lower coefficients relative to the highest one. The results are then normalized to C.sub.0 =1.0 which is valid since A and B can be appropriately scaled also.
Ions near .beta..sub.u =1.0 have a motion composed primarily of a pair of frequencies equally spaced on either side of 1/2f, the main r.f. frequency. There are also equally spaced pairs on either side of 11/2f, 21/2f, etc. but their contribution to the overall motion of the ion is less than ten percent. As the limit of .beta..sub.u =1 is approached, the coefficient C-2 approaches negative one, and the component frequencies associated with n=0 and n=-1 approach f/2. For example, consider a hypothetical ion in transit through a quadrupole field. If the a.sub.x and q.sub.x for the ion are 0.0000 and 0.9000 respectively, and if the frequency of the quadrupole field is 1,000,000 Hz, the relative magnitudes and the frequencies of the components of motion of the ion would be as tabulated below. The fact statement that a.sub.x is zero indicates that the quadrupole is being operated in the r.f.-only mode. As a result, the ion motion in the x and y dimensions have identical character as .beta..sub.x and .beta..sub.y are equal and the relative magnitudes of the C.sub.2n are the same.
______________________________________ f = 1000000 Hz q.sub.x = 0.9000 a.sub.x = 0.0 .beta..sub.x = .beta..sub.y = 0.915911 ______________________________________ n = -3 C.sub.-6 = -0.0027315 5f/2 + (1 - .beta..sub.x)f/2 = 2.542045 Mhz n = -2 C.sub.-4 = 0.0783999 3f/2 + (1 - .beta..sub.x)f/2 = 1.542045 Mhz n = -1 C.sub.-2 = -0.8258339 f/2 + (1 - .beta..sub.x)f/2 = 0.542045 Mhz n = 0 C.sub.+0 = 1.0000000 f/2 - (1 - .beta..sub.x)f/2 = 0.457955 Mhz n = 1 C.sub.+2 = -0.1062700 3f/2 - (1 - .beta..sub.x)f/2 = 1.457955 MHz n = 2 C.sub.+4 = 0.0039605 5f/2 - (1 - .beta. .sub.x)f/2 = 2.457955 MHz n = 3 C.sub.+6 = -0.0000745 7f/2 - (1 - .beta..sub.x)f/2 = 3.457955 MHz ______________________________________
When q.sub.x is closer to the stability limit, one finds the values for C.sub.0 and C.sub.-2 to be nearly equal in magnitude which indicates that the ion's trajectory is primarily a mixture of two sinusoidal components of nearly equal magnitude with frequencies very close to f/2. The relative magnitudes and frequencies of the two primary components of motion for this case are as tabulated below.
__________________________________________________________________________ q.sub.x = 0.907590 a.sub.x = 0.0 .beta..sub.x = .beta..sub.y = 0.980000 __________________________________________________________________________ n = 0 C.sub.+0 = 1.0000000 f/2 - (1 - .beta..sub.x)f/2 = 0.490000 MHz n = -1 C.sub.-2 = -0.9556011 f/2 + (1 - .beta..sub.x)f/2 = 0.510000 MHz __________________________________________________________________________
Such a trajectory is represented graphically in FIG. 5. The trajectory is plotted for the normalized time interval from .xi.=0 to 200, which is equivalent to 63.66 cycles of the frequency. This translates to 63.66 microseconds for this example. The two components of ion motion exhibit 31.19 and 32.46 cycles during this interval, a difference of 1.27 cycles. The composite trajectory appears as sinusoidal motion having a frequency of f/2, the average of the two frequencies associated with dominant components of the ion motion, undergoing beats. The frequency of these beats is difference between these same two component frequencies. When the q.sub.u for the ion is lower than ca. 0.4 only the coefficient corresponding to n=0 is significant, so the ion's motion is predominantly composed of a sinusoidal component of frequency .beta..sub.u .omega./2. This is illustrated in FIG. 6 which shows a trajectory for an ion having q.sub.u =0.2.
While the previous examples are for cases where there is no DC component of the quadrupole field, the illustrated dependence of the character of ion motion on the parameter .beta..sub.u is generally applicable to cases where a.sub.x is non zero. While a.sub.x and q.sub.x determine both .beta..sub.x and .beta..sub.y, .beta..sub.x and .beta..sub.y define the character of the motion in the x and y dimensions. Often in discussing the motion of ions, it is more descriptive, and therefore useful, to describe an ion in terms of its .beta. in a particular dimension than its corresponding a.sub.x and q.sub.x. When discussing motion in an r.f. only quadrupole, q.sub.x and .beta..sub.x are often used interchangeably as one uniquely defines the other. The relationship between q.sub.x and .beta..sub.x when a DC component to the quadrupole field is absent, i.e. a.sub.x =0, is of considerable importance to the discussion which follows. At low values of q.sub.x on the a.sub.x =0.0 axis one finds that a simple linear approximation is adequate to describe this relationship as shown in equation (15). ##EQU8## The change in .beta..sub.x with respect to q.sub.x may be found by differentiation to be approximately 0.7071 as shown in equation (16). ##EQU9## This approximate relationship holds up to about .beta..sub.x =0.4 where the slope begins to increase. It continues to increase asymptotically until, at the stability limit, where .beta..sub.x equals one and q.sub.x equals ca. 0.908, the slope is infinity. This means that for values for q.sub.x near the stability limit, a small change in q.sub.x will have a large effect upon .beta..sub.x and therefore the corresponding frequency components of the ion motion. This frequency dispersion is of fundamental importance to this invention.
Up to this point the discussion has dealt exclusively with the trajectories of ions in transit though a purely quadrupolar field. However, as will be described below, it can be useful to modify the potential field by adding small auxiliary field components having frequencies other than that of the main field. The most simple form of an auxiliary field is a dipole field. A dipolar potential field results in a electric field that is independent of displacement. The equations of motion for an ion in transit through such a perturbed quadrupole field have the form shown in equation (17). ##EQU10## This equation of motion is simply a forced version of the Mathieu equation. The term on the right hand side of the equation represents the additional component of force the ion is subject to in the dimension of interest, u, due to the dipolar auxiliary field. The parameter P.sub.u is proportional to the magnitude of the auxiliary electric field component in the u dimension. The parameters .alpha. and .theta..sub..alpha. represent the frequency, in normalized units, and the phase of the sinusoidally varying auxiliary field. The relationship between the unnormalized frequency of the auxiliary field, f.sub..alpha., and .alpha. is given in equation (18). The effect of this extra force term is strongly dependent upon the frequency of the auxiliary field. ##EQU11##
If this frequency, .alpha., matches any of the component frequencies of the ions' motion, (2n+.beta..sub.u), then a resonance condition exists. The general oscillatory character of the motion remains the same; however, the amplitude of the oscillatory motion grows linearly with time. The rate of growth of the amplitude of the ion's oscillatory motion is proportional to both the magnitude of the auxiliary field and the relative contribution of the component of unforced motion, as represented by C.sub.2n, in resonance with the applied field. Even though only one component of the ion's motion is in resonance with the auxiliary field, all components of the ions trajectory grow in concert thus maintaining their relative contribution to the trajectory.
When the frequency of the auxiliary field is only very close to one of the ion's resonant frequencies, the resultant ion oscillation beats with a frequency equal to the difference between the auxiliary field frequency and any nearby ion resonant frequencies. In the case where the auxiliary frequency corresponds to an .alpha. near unity, and the .beta..sub.u describing the ion's resonant frequencies in the field is near 1.0, the resultant trajectory has multiple beats as there are two resonant frequencies near the auxiliary frequency.
When the frequency of the auxiliary field is not close to any of the ion's resonant frequencies, the resultant ion oscillation is largely unaffected. Rigorous analysis shows that the presence of an auxiliary field always has some effect on an ion's trajectory, however, if the difference in frequency between the frequency of the auxiliary field, .alpha., and the closest ion resonant frequency, 2n+.beta..sub.u, is greater than a percent, .vertline..alpha.-(2n+.beta..sub.u).vertline./.beta..sub.u &gt;0.01, then the portion of the ion's trajectory due the auxiliary field will be negligible. The ion will essentially behave as if there were no auxiliary field present. This of course is assuming that the magnitude of the auxiliary field is relatively small.
So far we have discussed ion motion in the presence of a sinusoidally varying dipolar auxiliary field. Certainly, the auxiliary dipole field could vary in a more complicated way such that the right hand side of Equation (17) would become a generalized function of time, P.sub.u (.xi.), as is shown in equation (19). ##EQU12## Fourier theory says that if P.sub.u (.xi.) is periodic, then it can be expressed as an infinite series of sines and cosines having harmonic frequencies. Even if P.sub.u (.xi.) is not periodic it can be represented as an integral (a sort of sum) of sine and cosine terms having differentially spaced frequencies. Hence auxiliary fields having complicated time variance can be treated as the sum of multiple dipolar auxiliary fields, each varying sinusoidally and each having a different frequency. This results in a P.sub.u (.xi.) that is the sum of multiple cosine terms. Since the Mathieu equation is a linear differential equation, it has the useful property that superposition applies to its solutions. One can consider the trajectory described by Equation (19) as the sum of multiple independent trajectories, one accounting for ion motion in the absence of any auxiliary field, and other trajectories accounting for the motion associated with each frequency component of the auxiliary field. ##STR1##
For actual quadrupole mass filters, a dipolar auxiliary field can be created by symmetrically applying a differential voltage, 2U.sub.s (t), between opposing electrodes in addition to the common mode voltage, U(t) or -U(t), applied to both opposing electrodes in order to generate the main quadrupole field. For example, to establish an auxiliary dipole field oriented so that ions are only subject to an auxiliary force in the x dimension, one applies voltages U(t)+U.sub.s (t) and U(t)-U.sub.s (t) to the +x and -x electrodes, respectively, and a voltage of -U(t) to both the +y and -y electrodes. The resultant auxiliary potential field is predominately dipolar but it is not purely dipolar. FIG. 7 shows lines of equipotential in a cross section view of an auxiliary field applied to hyperbolic electrodes. Since FIG. 7 represents only the auxiliary portion of the potential field, the -x (left-hand) electrode has a potential of -U.sub.s (t), the +x (right-hand) electrode has a potential of U.sub.s (t) and both the +y and -y (upper and lower) electrodes have potentials of zero. For a purely dipolar field the equipotential lines would be parallel. In W. Paul's original patent, it is recognized that generating an auxiliary field in such a manner would not result in a purely dipole auxiliary field. The curvature of the equipotential lines in FIG. 7 is due to the higher order terms in a polynomial expansion that mathematically describe this auxiliary potential field. Equation (20) is a truncated polynomial expansion approximately describing the auxiliary potential field, .PHI..sub.s (x,y,t). This truncated expansion represents the auxiliary potential as a sum of first-order (dipole), third-order (hexapole), fifth-order (decapole) and seventh-order component fields. FIG. 7 was obtained using the computer program SIMION PC/AT which models potential fields using a grid relaxation technique. Equation (20) was obtained by a fit to the estimated potentials obtained by SIMION PC/AT. The dipole component of such an auxiliary field for the x dimension may be expressed as equation (21). ##EQU13## This equation may be substituted for the term p.sub.x (.xi.) in the normalized equation of motion for the x dimension version of equation (19). If the potentials applied to the +x and -x electrode were not applied symmetrically relative to both y electrodes, there would also be second-order (quadrupole) and, perhaps, if the electrodes had round rather than hyperbolic contours, higher even-order components of auxiliary field. It can be seen in equation (20) that the dipole and hexapole components account for most of the auxiliary field. The effect of the higher order components of the auxiliary field on the ion motion is a very difficult issue. The equations of motion that result when hexapole or higher order auxiliary field components are considered are nonlinear and coupled. Unlike the case of motion for ions in combined dipole and quadrupole fields, motion in the x dimension is effected by motion in the y dimension and vice versa. This means that x appears in the y dimension equation of motion and y appears in the x dimension equation of motion. This also means that there are terms in these equations of motion that are second order or greater. More specifically, there are terms of the form x.sup.a y.sup.b, where a+b is greater than unity, in the equations of motion. Even if only dipole and quadrupole components of the auxiliary field are considered in formulating the equations of motion, the resulting equations of motion are still very difficult to solve. All such equations are generally amenable only to numerical methods or approximation methods, such as perturbation methods, for their solution. Using perturbation methods it can be shown that higher order sinusoidally varying auxiliary field components cause resonances at frequencies, .alpha., other than those expected from the purely dipole auxiliary field model; however, these resonances are not nearly as strongly excited as a resonance excited by the main dipole component of the auxiliary field. If the magnitude of the auxiliary field is small enough so that, while in transit of the quadrupole, only ions having very narrow range of q.sub.x would have trajectories significantly altered by the presence of the dipole component of auxiliary field, then it is unlikely that the effect of the higher order components of the auxiliary field will have a significant effect on the trajectories of ions regardless of their q.sub.x.
If the magnitude of the auxiliary field is relatively large, resonances attributable the higher order auxiliary field components can be significant. These effects have been observed experimentally.
Theoretically it possible to create an auxiliary potential that is primarily composed of higher order components. However, this would most likely involve altering the design of the quadrupole electrode structure which would compromise purity of the main quadrupole field. The one exception to this is that one can apply a very pure quadrupole auxiliary field simply by adding a different frequency component to the voltage applied between electrode pairs. A well known resonance associated with quadrupolar auxiliary fields is defined in Equation (22). EQU .alpha.=2.beta..sub.u ( 22)
The disadvantage of using a quadrupole auxiliary field is that resonances will occur in both the x and y dimension simultaneously. A dipole field can be oriented, as is the one described above, so as to cause resonance in only a single dimension of motion.
Usually, when a linear quadrupole field is used as a mass filter, both r.f. and DC voltages are employed. In this case, the apex of the first stability region is cut by a line, representing the locus of all possible masses, which passes through this apex as seen in FIG. 4. Mass is inversely related to q.sub.x. For an arbitrary r.f. voltage, U.sub.r.f., ion masses can be thought of as points which are spaced inversely with mass along the line such that infinite mass is at the origin and mass zero is at infinity. This is known as the scan line. The range of masses which map within the stability region along this line defines the mass range that will pass the mass filter. The slope of this line is determined by the ratio of the r.f. and DC voltages, which sets the resolution by limiting the minimum and maximum q values that permit an ion to pass. The range of q.sub.x that corresponds to the portion of a given scan line that is within the stability region is sometimes referred to as the transmission band. Proper choice of the r.f./DC ratio allows only one ion mass to pass at a time. To obtain a mass spectrum, the r.f. and DC voltages are increased. The position of ion masses on the scan line shift away from the origin, bringing successive ion masses through the narrow tip of the stability region. Higher masses are spaced closer on the line, so, to maintain unit mass resolution, the slope of the line must increase as mass increases. A plot of ion current detected at the quadrupole exit versus the applied voltage is a mass spectrum.
A convenient way to visualize this process is to imagine the scan line as an elastic string, with one end fixed to the origin of the stability diagram. Individual masses are represented as points marked on the string. The spacing is inversely proportional to mass, therefore, the spacing is closer towards the origin where higher masses are found than it is at the low mass end of the string. Increasing the amplitude of the r.f. and DC voltages has the effect of stretching the string. As the string is stretched, the slope is increased gradually so that only one mark falls within the stability region at a time.
There are several problems with this mode of operation. The most severe is the ion transmission penalty encountered as resolution is increased at high mass. A second problem is the sensitivity to contamination, primarily due to charge accumulation, which distorts the quadrupole fields. Operation modes involving r.f.-only fields have been proposed to overcome these deficiencies.
The simplest r.f.-only mass filter uses the high q.sub.x cutoff to provide a high pass filter. At a given r.f. voltage setting, the higher masses, which have q.sub.x s lower than 0.908, have stable trajectories while the lower masses, which have q.sub.x s above 0.908 will have unstable trajectories. A scan of the r.f. voltage from low amplitude to high amplitude while detecting the ion current exiting the mass filter produces a plot of detected ion current versus voltage that resembles a series of decreasing stair steps. These steps, in general, have neither consistent spacing nor height. The r.f. voltage at which each step occurs corresponds to the passing from stability to instability of a particular ion mass present in the ion beam. The magnitude of each vertical transition is proportional to the abundance of a corresponding ion mass present in the ion beam injected into the mass filter. The first derivative of this curve is a mass spectrum.
There are several problems with the straightforward approach to converting the measured ion current versus r.f. voltage stair step function to a mass spectrum. Due to the statistical variation inherent in the rate of ion arrival at any ion detector, there is a noise component associated with any detected ion current signal. This noise, is essentially white as it has a uniform power spectrum. The magnitude of this ion statistical noise is proportional to the square root of the average intensity of the detected ion current signal. Small mass peaks are seen in the undifferentiated ion signal as small steps on a large offset produced by the transmission of all higher masses. The process of differentiation enhances the high frequency components of the signal relative to the low frequency components. The ion statistical noise accompanying this large ion signal offset when enhanced by differentiation interferes with the observation of small mass peaks.
For well-constructed quadrupoles there can be an anomalous peak associated with the small stability region near a.sub.x,q.sub.x =0.0,7.5 in which ions of lower mass are stable. If the mass filter is operated as a broad band mass filter, but not r.f. only, the ratio of a.sub.x to q.sub.x can be maintained such that the artifact signal associated with this higher stability region can be avoided.
Another problem is the variation of ion transit probability within the transmission band. Any change in ion transmission as a function of q.sub.x, and therefore the r.f. voltage, will induce a response. Genuine mass peaks are difficult to distinguish in the presence of this uncorrelated response.
The r.f.-only operation mode can only be useful as a mass spectrometer if the stair step in the detected ion current to r.f. voltage function can be converted into mass peaks without amplifying the noise. Several ways to do this have been proposed and reduced to practice. In U.S. Pat. No. 4,090,075 granted in 1972, U. Brinkman disclosed a method for overcoming some of the limitations outlined above. As ions become unstable at high q.sub.x, they take on a large transverse kinetic energy. In the fringing fields at the exit of the quadrupole, the large radial excursions subject the ions to intense axial fields, thereby causing them to acquire large axial kinetic energies. Placement of a retarding grid between the exit and the ion detector forms a coarse kinetic energy filter, which only passes those ions near the stability limit.
Another method, which takes advantage of the exit characteristics of ions near a stability limit, uses an annular detector which is described by J. H. Leck in British Patent 1,539,607. This scheme uses a central stop, biased to attract ions with low radial energies. Ions that possess enough transverse energy to avoid the central stop are collected on a ring that surrounds the central electrode.
In U.S. Pat. No. 4,189,640 granted Feb. 19, 1980, P. H. Dawson presents an alternative annular design that uses grids. The first grid is placed immediately following the r.f.-only quadrupole exit and is strongly biased to attract ions. A central stop is fixed to the grid to block axial ions from passing, and a second grid is placed to decelerate the ion beam. Ions of interest can then pass to a detector placed after the grids.
All of these techniques share the common strategy for reducing both major noise sources. They all attempt to detect only ion currents carried by ion masses that are very near the transition from stability to instability. This minimizes the ion statistical noise signal and thus improves detection limits. Although impressive results at low mass have been shown in the literature, attempts to apply these methods at higher mass have shown mass dependant leading edge liftoff which restricts their usefulness.
Modulation techniques have also been employed to convert the r.f.-only ion intensity function into mass peaks. The method involves encoding the component of the ion current signal corresponding to an ion mass at the stability threshold with a specific frequency and then using phase sensitive detection to monitor only that frequency. This eliminates the need to perform differentiation to obtain a mass spectrum. Coherent noise that falls outside the bandpass of the filter used in the detection system is discriminated against, thereby improving the signal to noise ratio.
This methodology was first used for r.f.-only mass spectrometry by H. E. Weaver and G. E. Mathers in 1978 (Dynamic Mass Spectrometry 5 (1987) pp 41-54). Their technique modulates the amplitude of the r.f. voltage at a specific frequency. The amplitude of the modulation of the r.f. voltage is a very small percentage of the average amplitude of the r.f. voltage. When the average r.f. amplitude is such that a particular ion mass has an average q.sub.x that approaches to the high q.sub.x stability limit, this threshold mass is brought in and out of stability at the same frequency as the amplitude modulation of the r.f. voltage. The modulation of the r.f. voltage thus alternately allows and prevents the transit of ions having the threshold mass through the mass filter to the detector. The component of the detected ion current carried by ions having this threshold mass is thus converted into an AC signal having frequency components equal to the frequency of the r.f. voltage amplitude modulation and its harmonics.
P. H. Dawson U.S. Pat. No. 4,721,854) presents a similar idea in which the DC component of the quadrupole field is modulated rather than the r.f. component of the quadrupole field. In this approach, ion stability is modulated by varying the stability parameter a.sub.x. By changing the amplitude of a small DC quadrupole voltage at a frequency which is low compared to the ion flight time through the quadrupole, the transmission of an ion mass corresponding to values of q.sub.x,a.sub.x near the .beta..sub.x =1 or .beta..sub.y =1 stability limit is modulated.
These modulation methods suffer from the substantial deficiency that the means used to modulate the current carried by ions having the mass of interest also weakly modulates the current carried by higher mass ions having corresponding a.sub.x s and q.sub.x s that are well within the stability region. This occurs because there is some variation in ion transmission with a.sub.x and q.sub.x throughout the stability region. The modulated ion current associated with ion masses not at the stability threshold is effectively a noise signal. Neither of these techniques, when implemented as true r.f. or AC only techniques, avoid the generation of artifact peaks associated with the small stability region at a.sub.x =0, q.sub.x =7.5.
The idea of using resonance excitation with r.f.-only quadrupoles is not new. In 1958 W. Paul, et al. (Zeitschrift fur Physik 164, 581-587 (1961) and 152, 143-182 (1958) described an isotope separator that uses an auxiliary dipole AC field to excite the oscillatory motion of an ion contained within an r.f.-only quadrupole field. This mass filter is operated so that the isotopes of interest are near the center of the stability diagram, such as near (a=0.0, q-0.6). This r.f.-only field will have no mass separation capability for the isotopes but the ion transmission will be very good. The auxiliary AC field is tuned to the fundamental frequency of ion motion for a specific isotopic mass. When this auxiliary field is included, ions of the selected mass will absorb energy and their amplitude of oscillation will increase. The trajectories of ions of nearby masses will also be affected. Their amplitude of oscillation will be modulated at a beat frequency equal to the difference between the excitation frequency and their frequency of motion. If the envelope of this amplitude modulation is greater than r.sub.o, the ion will not be transmitted. The auxiliary field can be either a quadrupole field applied at twice the frequency of ion motion, or it can be a dipole field applied at the frequency of ion motion. This excitation forms a basis for mass separation by eliminating one or a group of isotopes while permitting the desired isotope to be transmitted; however, this method relies on knowledge of the distribution of ion masses in the ion beam being injected into the quadrupole mass filter.
The use of auxiliary quadrupole and dipole resonance fields to add energy to an ion or electron beam is also discussed in detail with an excellent gravitational model in U.S. Pat. No. 3,147,445 by R. F. Wuerker and R. V. Langmuir, granted Sept. 1, 1964. That patent covers many applications of r.f.-only quadrupoles to manipulate ion or electron beams for electronic signal conditioning applications.
In U.S. Pat. No. 3,321,623 granted May 23, 1967 to W. M. Brubaker and C. F. Robinson, it is claimed that an auxiliary dipole field enhances the effectiveness of a quadrupole field by forcing ions from the axis to a larger radial displacement, where the quadrupole field has a greater effect. In practice, however, it can be shown that an oscillating dipole field of sufficiently small magnitude will have no noticeable effect unless its frequency is close to a frequency of the ion's natural motion.